Since the rarefaction waves \( (v^r_\pm , u^r_\pm , \theta ^r_\pm )(\frac{x}{t}) \)  are not smooth enough solutions, we shall construct smooth approximate ones. Motivated by [1]}, the smooth
solutions of Euler system (REF ), \( (V^r_\pm , U^r_\pm , \Theta ^r_\pm )(x,t) \) , which approximate \( (v^r_\pm , u^r_\pm , \theta ^r_\pm )(\frac{x}{t}) \) , are given by
\(\left\lbrace \begin{aligned}&\lambda _{\pm }\left(V_{\pm }^{r}(x, t), s_{\pm }\right) = w_{\pm }(x, 1+t) , \\[2mm]&U_{\pm }^{r}(x, t) = u_{\pm }-\int _{v_{\pm }}^{V_{\pm }^{r}(x, t)} \lambda _{\pm }\left(\eta , s_{\pm }\right) \mathrm {d} \eta , \\[2mm]&\Theta _{\pm }^{r}(x, t) = \theta _{\pm } v_{\pm }^{\gamma -1} \left( V_{\pm }^{r}(x, t) \right)^{1-\gamma } ,\end{aligned}\right.\) 
